A die is rolled a certain number of times.

Question

A die is rolled a certain number of times. If $P(	ext{odd twice})=P(	ext{even 3 times})$, find $P(	ext{odd number of times an odd number of times occurs})$.

Accepted Answer

${}^nC_2\left(\dfrac12\right)^n={}^nC_3\left(\dfrac12\right)^n\Rightarrow n=5$.

$$P(\text{odd count of "odd" outcomes}) = {}^5C_1\left(\frac12\right)^5+{}^5C_3\left(\frac12\right)^5+{}^5C_5\left(\frac12\right)^5 = \frac{5+10+1}{32}=\frac12$$

Solutions · 1